A272387 -id:A272387 - cp's OEIS frontend

A256891 Smallest primes of 3 X 3 magic squares formed from consecutive primes.

1480028129, 1850590057, 5196185947, 5601567187, 5757284497, 6048371029, 6151077269, 9517122259, 19052235847, 20477868319, 23813359613, 24026890159, 26748150199, 28519991387, 34821326119, 44420969909, 49285771679, 73827799009, 73974781889, 74220519319, 76483907837, 76560277009, 80143089599, 85892025227, 89132925737, 95515449037, 99977424653

Offset: 1

Arkadiusz Wesolowski, Apr 12 2015

nonn

Let a = a(n) for some n and {a, b, c, d, e, f, g, h, i} be the set of consecutive primes. Then it is:
   +---+---+---+              +---+---+---+
   | d | c | h |              | c | d | h |
   +---+---+---+              +---+---+---+
   | i | e | a | (type 1), or | i | e | a | (type 2). See Harvey D. Heinz.
   +---+---+---+              +---+---+---+
   | b | g | f |              | b | f | g |
   +---+---+---+              +---+---+---+
  The type is determined by the sign of A343195.
For a given magic sum S, it is easy to calculate the unique set of n^2 consecutive primes that sum up to n*S (see PROGRAM MagicPrimes() in A073519), and in particular the smallest of these (cf. PROGRAM), listed here for n = 3, in A260673 for n = 4, in A272386 for n = 5, and in A272387 for n = 6. - M. F. Hasler, Oct 28 2018

Allan W. Johnson, Jr., Consecutive-Prime Magic Squares, Journal of Recreational Mathematics, vol. 15, 1982-83, pp. 17-18.
H. L. Nelson, A Consecutive Prime 3 x 3 Magic Square, Journal of Recreational Mathematics, vol. 20:3, 1988, p. 214.

A.H.M. Smeets, Table of n, a(n) for n = 1..759
Harvey D. Heinz, Prime Numbers Magic Squares: Minimum consecutive primes - 3, 1999-2010.
Eric Weisstein's World of Mathematics, Prime Magic Square
Index entries for sequences related to magic squares

Cf. A073519, A151799, A166113, A260673, A272386, A272387, A343194, A343195.

Subsequence of A265139.

/* Brute-force search */ lst:=[]; n:=3; while n lt 10^11 do a:=NextPrime(n); q:=a; j:=a-n; if j mod 6 eq 0 then b:=NextPrime(a); if j eq b-a then c:=NextPrime(b); d:=c-b; if d mod 6 eq 0 then e:=NextPrime(c); k:=e-c; if k eq j then f:=NextPrime(e); if k eq f-e then g:=NextPrime(f); if g-f eq d then h:=NextPrime(g); m:=h-g; if m eq k then i:=NextPrime(h); if h-g eq i-h then Append(~lst, n); end if; end if; end if; end if; end if; end if; end if; end if; n:=q; end while; lst;

A256891(n)=MagicPrimes(A270305(n),3)[1] \\ See A073519 for MagicPrimes(). - M. F. Hasler, Oct 28 2018

a(n) = A151799(A151799(A151799(A151799(A166113(n))))). - Max Alekseyev, Nov 02 2015

Extended by Max Alekseyev, Nov 02 2015

A260673 Smallest primes of 4 X 4 magic squares formed from consecutive primes.

Original entry on oeis.org

31, 37, 1229, 4931, 12553, 3259909, 3324329, 9291521, 24066643, 26025107, 46330021, 95979511, 99268649, 116923057, 170995151, 204041417, 213084871, 218568971, 229981399, 232850557, 254042641, 255432869, 256714219, 300222341, 375303157, 383432249, 421514827

Offset: 1

Arkadiusz Wesolowski, Nov 14 2015

nonn

			        n = 3
|----|----|----|----|
|1229|1249|1321|1319|
|----|----|----|----|
|1301|1303|1231|1283|
|----|----|----|----|
|1297|1277|1307|1237|
|----|----|----|----|
|1291|1289|1259|1279|
|----|----|----|----|
.
        n = 4
|----|----|----|----|
|4943|4933|5011|5009|
|----|----|----|----|
|4999|4973|4967|4957|
|----|----|----|----|
|5003|4969|4987|4937|
|----|----|----|----|
|4951|5021|4931|4993|
|----|----|----|----|

Allan W. Johnson, Jr., Consecutive-Prime Magic Squares, Journal of Recreational Mathematics, vol. 15, 1982-83, pp. 17-18.

Zhao Hui Du, Table of n, a(n) for n = 1..1000
Harvey D. Heinz, Prime Numbers Magic Squares: Minimum consecutive primes - 4, 1999-2010.
Carlos Rivera, Puzzle 3. Magic Squares with consecutive primes, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Prime Magic Square
Index entries for sequences related to magic squares

Cf. A073521, A173981, A256891, A270864, A272386 (analog for n=5), A176571 (magic sums for n=5), A272387. Subsequence of A270865.

A260673(n)=MagicPrimes(A173981(n),4)[1] \\ See A073519 for MagicPrimes(). - M. F. Hasler, Oct 28 2018

Extended by Arkadiusz Wesolowski, Dec 13 2015

A177434 The magic constants of 6 X 6 magic squares composed of consecutive primes.

Original entry on oeis.org

484, 744, 806, 868, 930, 1390, 1460, 1494, 1634, 1704, 1740, 1848, 1992, 2100, 2172, 2316, 2390, 2540, 3116, 3192, 3694, 3734, 3774, 4486, 4946, 4988, 5736, 6104, 6148, 6526, 6568, 6610, 6776, 6820, 6950, 7036, 7078, 7120, 7984, 8118, 8162, 8828, 9318

Offset: 1

Natalia Makarova, May 08 2010

nonn

Let Z be a sum of 36 consecutive primes. A necessary condition to get a 6 X 6 magic square using these primes is that Z=6S, where S is even. The smallest magic constant of a 6 X 6 magic square of consecutive primes is 484 (cf. A073520).
Each of the first 100 possible arrays of 36 consecutive primes which satisfy the necessary condition produces a magic square.
A program written by Stefano Tognon was used.

			S = 744
   [139 113 151 131  83 127]
   [223 149  89  47 157  79]
   [173 103 181 167  59  61]
   [ 67 137  53  97 211 179]
   [101 199  73 109  71 191]
   [ 41  43 197 193 163 107]
S = 806
   [131  53 107 157 191 167]
   [ 89 229 179  97 109 103]
   [ 83 211  71 139  79 223]
   [113 101 137 181 227  47]
   [197  61 163  59 127 199]
   [193 151 149 173  73  67]
S = 868
   [191 137  79 193 197  71]
   [ 67 157  73 229 239 103]
   [179 173 167  97 101 151]
   [211 181 223  61 109  83]
   [113 131 199 139  59 227]
   [107  89 127 149 163 233]
Magic square with S=930 can be pan-diagonal (cf. A073523).
Example of a non-pan-diagonal square:
S = 930
   [167  71 151 199 131 211]
   [ 89 241 181  73 113 233]
   [ 83 227 127 197 229  67]
   [239 137 139 103 163 149]
   [179  97 223 251 101  79]
   [173 157 109 107 193 191]

Natalya Makarova, Author's webpage (in Russian)

Cf. A173981 (analog for 4 X 4), A176571 (analog for 5 X 5), A073523 (36 consecutive primes of a pandiagonal magic square), A073520 (smallest magic sum for n X n), A259733 (most-perfect 8 X 8), A272387 (smallest element of 6 X 6 magic squares of consecutive primes).

A177434(n, p=A272387[n], N=6)=sum(i=2, N^2, p=nextprime(p+1), p)/N \\ Uses a precomputed array A272387, but can actually be used to find the terms, cf A272387. - M. F. Hasler, Oct 28 2018

a(n) = Sum_{k=0..35} A000040(A000720(A272387(n))+k)/6. - M. F. Hasler, Oct 28 2018

Edited by M. F. Hasler, Oct 28 2018

A272386 Smallest primes of 5 X 5 magic squares formed from consecutive primes.

Original entry on oeis.org

13, 59, 79, 97, 107, 127, 157, 269, 337, 347, 439, 457, 479, 563, 601, 631, 719, 743, 883, 947, 1021, 1031, 1049, 1051, 1061, 1093, 1109, 1171, 1201, 1223, 1499, 1523, 1601, 1669, 1811, 1901, 1933, 1997, 2011, 2053, 2153, 2207, 2341, 2399, 2531, 2539, 2549, 2551

Offset: 1

Arkadiusz Wesolowski, Apr 28 2016

nonn

A necessary condition for a prime being in this sequence is that the sum of this and the subsequent 24 primes divided by 5 must be an odd integer. - M. F. Hasler, Oct 30 2018

			The smallest 5 X 5 magic square that can be formed from 25 consecutive primes consists of the primes 13 through 113, so the first term is 13:
           n = 1
|----|----|----|----|----|
| 13 | 107| 73 | 101| 19 |
|----|----|----|----|----|
| 97 | 17 | 79 | 37 | 83 |
|----|----|----|----|----|
| 41 | 53 | 109| 43 | 67 |
|----|----|----|----|----|
| 103| 89 | 29 | 61 | 31 |
|----|----|----|----|----|
| 59 | 47 | 23 | 71 | 113|
|----|----|----|----|----|
The next smallest consists of the primes 59 through 179, so the second term is 59:
          n = 2
|----|----|----|----|----|
| 59 | 163| 151| 137| 67 |
|----|----|----|----|----|
| 149| 61 | 79 | 109| 179|
|----|----|----|----|----|
| 113| 83 | 173| 107| 101|
|----|----|----|----|----|
| 167| 139| 71 | 127| 73 |
|----|----|----|----|----|
| 89 | 131| 103| 97 | 157|
|----|----|----|----|----|

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..66
Eric Weisstein's World of Mathematics, Prime Magic Square
Arkadiusz Wesolowski, Examples of these magic squares
Index entries for sequences related to magic squares

Cf. A176571, A256891, A260673, A272387.

A272386(n)=MagicPrimes(A176571(n),5)[1] \\ See A073519 for MagicPrimes(). - M. F. Hasler, Oct 28 2018

is_candidate(p)={denominator(p=A173981(,p))==1 && bittest(p,0)} \\ For p < 167, this yields exactly the terms of A272386. Exceptions (primes satisfying this but not in A272386) are (167, 227, 383, 461, 607, ...). - M. F. Hasler, Oct 30 2018

cp's OEIS Frontend

A256891 Smallest primes of 3 X 3 magic squares formed from consecutive primes.

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A260673 Smallest primes of 4 X 4 magic squares formed from consecutive primes.

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A177434 The magic constants of 6 X 6 magic squares composed of consecutive primes.

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A272386 Smallest primes of 5 X 5 magic squares formed from consecutive primes.

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